50 MATHCOUNTS LECTURES (1) LOGICAL REASONING BASIC ...

50 MATHCOUNTS LECTURES (1) LOGICAL REASONING
Example: A jeweler has four small bars that are supposed to be gold. He knows
that one is counterfeit and the other three are genuine. The counterfeit bar has a
slightly different weight than a real gold bar. Using a balance scale, what is the
minimum number of weighings necessary to guarantee that the counterfeit bar
will be detected? (Mathcounts)
Solution: 2.
We divide the four bars into three groups: 1, 1, and 2. We weight two bars, say,
bar A and bar B, first.
Case I: If their weights are different, we remove one, say, bar A, and put a third
bar, say bar C. If B and C are the same, and then bar A is the counterfeit. If bar B
and bar C are different, bar B is the counterfeit (since it’s weight is different from
both A and C).
Case II: If their weights are the same, then we remove one, say, Bar A, and put a
third bar, say Bar C. If B and C are the same, then Bar D is the counterfeit. If Bar
B and Bar C are different, Bar C is the counterfeit.
So two weighings are necessary.
(e). Drawing Solid and Dash Lines
Example: Three friends – math teacher Mr. White, science teacher Mr. Black,
and history teacher Mr. Redhead – met in a cafeteria. “It is interesting that one of
us has white hair, another one has black hair, and the third has red hair, though no
one’s name gives the color of their hair” said the black-haired person. “You are
right,” answered White. What color is the history teacher’s hair?
Solution: The history teacher’s hair is black.
If the relationship of two things is certain, we draw a solid line between them.
Otherwise, we draw a dash line.
We know that no one’s name gives the color of
their hair. So we draw the dash lines as shown on
the right:
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